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Let’s Count the Ways
Overview – Activity ID: 8938
Math Concepts
Students will be introduced to the different ways to calculate
numbers of outcomes, including using the counting principle.
They will also evaluate expressions using permutations of data
without repeat elements, both manually and on the TI-34
MultiView Calculator.
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counting principle
fractions
factorials
permutations
Materials
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TI-34
MultiView™
pencil
paper
Activity
Give the students an example of when they might need the multiplication counting principal.
You are trying to decide what you want for lunch in the cafeteria today, and you feel overwhelmed by
the options. You decide upon a sandwich, but still, there are many choices. You can select wheat or
white bread, then ham, turkey, or roast beef. How many choices do you really have if you’re allowed
one type of meat?
Most students will create sketches of some kind to come up with their answers. Here is the beginning
of one example of what students might attempt. This is called a “tree diagram.”
wheat
ham
white
turkey
roast beef
You can count the possibilities using a diagram. On wheat, you can choose ham, turkey, or roast beef.
So there are three possible sandwiches on wheat bread. There are also three possible sandwiches on
white bread since white can be paired with ham, turkey, or roast beef, also. We can see there are six
total possibilities.
Now, what if we had the option of four different types of bread and five types of meat? If offered
white, wheat, rye, or multigrain bread, then ham, turkey, roast beef, chicken, or corned beef, how
many sandwich possibilities would there be (still allowing one type of meat per sandwich)? A
diagram similar to the one we just used is still possible, but the more options we have, the more
complicated the diagram becomes.
Introduce the counting principle.
When determining the number of possible outcomes, multiply the number of ways each separate
possibility can occur.
For instance, in the initial example, there were two types of bread and three types of meat. Multiply
2 • 3 to find that there are six possible sandwich combinations. In the next example, there were four
types of bread and five types of meat. Multiply 4 • 5 to see that there are 20 possible combinations.
In both previous examples, we were simply counting numbers of possibilities. Next, introduce the
concept of permutations before giving a formal definition.
In both sandwich problems, we were simply counting possibilities. There was no need to consider
order. However, sometimes order is important. For instance, consider this example: A track coach is
trying to decide the order of his sprinters for the 4 × 100-meter relay. How many possibilities are
there?
©2008 Texas Instruments Incorporated
TI-34 MultiView™ – Activity – p. 1 of 3
Let’s Count the Ways
Discuss the relevance of order. Even though each runner
runs one leg, the order “Josh, Jake, Lucas, Cody” is
different from “Josh, Jake, Cody, Lucas,” etc.
There are four choices for the runners of the first leg; then
for the second leg, there are three options remaining since
one runner is already chosen to run the first leg.
Continuing, there are two options for the third leg, then one
runner left for the last leg.
(4 runners) • (3 runners) • (2 runners) • (1 runner) = 24
This is a "factorial." The factorial n! is defined as the
product of all positive integers in descending order from n
to 1. For example, 4! = 4 • 3 • 2 • 1.
Show how to use the calculator to calculate a factorial.
Now, move on to examples in which not all objects are
used.
The above example is also called a "permutation." Any
particular arrangement of the elements of a set is a
permutation. But not all permutations use all the objects or
elements. What if the track coach in the previous example
were trying to decide which sprinters should run the 4 ×
100-meter relay, but instead of having just four runners, he
had nine possible runners to fill those four spots?
Follow these steps:
1. Press 4.
2.
Press H $ $ to access the
factorial command.
3.
The calculator screen should
display this:
4.
Press <
<
.
Start again with the counting principle.
If only three of the boys were good starters, the first leg
could be run by any of those three runners. Then the second
leg could be filled by any of the eight remaining runners. If
we continue this pattern, we will find that
(3 runners) • (8 runners) • (7 runners) • (6 runners)
= 1,008.
©2008 Texas Instruments Incorporated
TI-34 MultiView™ – Activity – p. 2 of 3
Let’s Count the Ways
Discuss how this is completely different from 9!, as 9!
would be 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1, and the above is
3 • 8 • 7 • 6.
We define the number of permutations of r objects taken
from a set of n objects as follows:
n Pr
=
n!
( n − r )!
In the example of nine runners and four possible slots to
fill, n = 9 and r = 4. Therefore, we could say this:
9P4=
9!
(9 - 4)!
=
9!
5!
=
9 i 8 i 7 i 6 i 5i 4 i 3i 2 i 1
5i 4 i 3i 2 i 1
Show the students on a chalkboard or overhead projector
how to simplify this fraction first, then multiply what
remains. The answer will be 3,024, just as before.
Now that you’ve seen how large the numbers can get and
how many factors there are, you can see why using a
calculator can simplify the process. Let’s use the TI-34
MultiView.
Do one more example with the students, again stressing that
order is important and reminding them we are strictly
dealing with problems that have no repeat elements.
Follow these steps:
1.
Press 9.
2.
Press H <
to access the
permutations command.
3.
The calculator screen should
display this:
4.
Press 4 <
.
Let’s try one more example. How many four-letter
arrangements can be made from the letters in the word
"movies"? The four-letter arrangements do not need to be
words, so "movs" qualifies as a legitimate arrangement.
Follow these steps:
2.
Press ! ! ! 7 <
We have seven letters and want arrangements of four
letters. So, for the first letter there are seven choices, for
the second letter there are six choices, etc. Using our
formula for permutations, we find this:
3.
The calculator screen should
display this:
7P4=
7!
(7 - 4)!
=
7!
3!
=
1. Press # # <
to access and
copy the previous entry.
.
7 i 6 i 5 i 4 i 3i 2 i 1
3i 2 i 1
We could use the counting principal: 7 • 6 • 5 • 4,
or on the calculator, we can more easily find 7P4 = 840.
©2008 Texas Instruments Incorporated
TI-34 MultiView™ – Activity – p. 3 of 3
Name ______________________
Let’s Count the
Ways
Date
______________________
Directions: Use the TI-34 MultiView™ to evaluate expressions using permutations and factorials.
For problems 1-6, evaluate or simplify. To demonstrate understanding, you must first write out the
factors. Then you may use your calculator. Problem 1 has been started for you to demonstrate.
1. 8!
2. 4! + 5!
8•7•6•5•4•3•2•1=
3. 7!
5.
9!
3!
4. 3(6!)
6.
10!
5!
For problems 7 and 8, determine the number of permutations in each given situation. Notice that in both
situations, repeats are not allowed.
7. The student council is in the process of electing officers. How many ways can a president, vicepresident, secretary, and treasurer be selected if there are 19 members in the group?
8. You would like a cone with two scoops of sherbet and are wondering how many different ways there
are to use two different flavors to create your own distinct taste. The possibilities are cherry, grape,
lime, watermelon, orange, blueberry, and pineapple. A scoop of blueberry on the bottom, then a scoop
of orange on top tastes different than if the scoop of orange is on the bottom. Find the number of twoflavor possibilities.
©2008 Texas Instruments Incorporated
TI-34 MultiView™ – Worksheet – Page 1 of 2
Let’s Count the
Ways
Name ______________________
Date
______________________
In problems 9-11, use any method you prefer to find the number of different arrangements in each given
situation.
9. It’s time for you to set your class schedule for next fall. If you need to take English, French, U.S.
history, Art II, geometry, and weight lifting and you have a six-period day, in how many different
orders could you take your classes?
10. Each locker in your school has a 10-digit lock with the numbers 1 through 10 each appearing once.
How many different 3-digit arrangements are possible if no single number is to be used twice on the
same locker?
11. You are about to move into your first home! You have narrowed down your color scheme, but you're
having trouble deciding which colors to use and want to know how many options you’re considering.
Your carpet options are tan or white; your paint colors are warm muffin, Aztec brick, almond brown,
and stone brown; and you also need to decide between chrome or brass hardware. How many total
possibilities are you considering?
©2008 Texas Instruments Incorporated
TI-34 MultiView™ – Worksheet – Page 2 of 2
Let’s Count the Ways
ANSWER KEY
1. 8!
2. 4! + 5!
8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320
3. 7!
4 • 3 • 2 • 1 + 5 • 4 • 3 • 2 • 1 = 24 + 120 = 144
4. 3(6!)
7 • 6 • 5 • 4 • 3 • 2 • 1 = 5,040
5.
9! 9 i 8 i 7 i 6 i 5 i 4 i 3 i 2 i1
=
= 60,480
3!
3 i 2 i1
3(6 • 5 • 4 • 3 • 2 • 1) = 2,160
6.
10! 10 i 9 i 8 i 7 i 6 i 5 i 4 i 3 i 2 i1
=
= 30,240
5!
5 i 4 i 3 i 2 i1
7. The student council is in the process of electing officers. How many ways can a president,
vice-president, secretary, and treasurer be selected if there are 19 members in the group?
19P4 =
19!
= 93,024
(19 − 4)!
8. You would like a cone with two scoops of sherbet and are wondering how many different ways
there are to use two different flavors to create your own distinct taste. The possibilities are
cherry, grape, lime, watermelon, orange, blueberry, and pineapple. A scoop of blueberry on the
bottom, then a scoop of orange on top tastes different than if the scoop of orange is on the
bottom. Find the number of two-flavor possibilities.
7P2 =
7!
= 42
(7 − 2)!
9. It’s time for you to set your class schedule for next fall. If you need to take English, French,
U.S. history, Art II, geometry, and weight lifting and you have a six-period day, in how many
different orders could you take your classes?
6! = 720
10. Each locker in your school has a 10-digit lock with the numbers 1 through 10 each appearing
once. How many different 3-digit arrangements are possible if no single number is to be used
twice on the same locker?
10P3=
720
11. You are about to move into your first home! You have narrowed down your color scheme, but
you're having trouble deciding which colors to use and want to know how many options you’re
considering. Your carpet options are tan or white; your paint colors are warm muffin, Aztec
brick, almond brown, and stone brown; and you also need to decide between chrome or brass
hardware. How many total possibilities are you considering?
2 • 4 • 2 = 16
©2008 Texas Instruments Incorporated
TI-34 MultiView™ – Worksheet – Answer Key